4 research outputs found
Cooperative Data Exchange based on MDS Codes
The cooperative data exchange problem is studied for the fully connected
network. In this problem, each node initially only possesses a subset of the
packets making up the file. Nodes make broadcast transmissions that are
received by all other nodes. The goal is for each node to recover the full
file. In this paper, we present a polynomial-time deterministic algorithm to
compute the optimal (i.e., minimal) number of required broadcast transmissions
and to determine the precise transmissions to be made by the nodes. A
particular feature of our approach is that {\it each} of the
transmissions is a linear combination of {\it exactly} packets, and we
show how to optimally choose the value of We also show how the
coefficients of these linear combinations can be chosen by leveraging a
connection to Maximum Distance Separable (MDS) codes. Moreover, we show that
our method can be used to solve cooperative data exchange problems with
weighted cost as well as the so-called successive local omniscience problem.Comment: 21 pages, 1 figur
Generalized Reed-Solomon Codes with Sparsest and Balanced Generator Matrices
We prove that for any positive integers and such that , there exists an generalized Reed-Solomon (GRS) code that
has a sparsest and balanced generator matrix (SBGM) over any finite field of
size , where sparsest means that
each row of the generator matrix has the least possible number of nonzeros,
while balanced means that the number of nonzeros in any two columns differ by
at most one. Previous work by Dau et al (ISIT'13) showed that there always
exists an MDS code that has an SBGM over any finite field of size , and Halbawi et al (ISIT'16, ITW'16) showed that there exists
a cyclic Reed-Solomon code (i.e., ) with an SBGM for any prime power
. Hence, this work extends both of the previous results
Cooperative Data Exchange with Weighted Cost based on d-Basis Construction
We consider the cooperative data exchange problem, in which nodes are fully connected with each other. Each node initially only has a subset of the K packets making up a file and wants to recover the whole file. Node i can make a broadcast transmission, which incurs cost w_i and is received by all other nodes. The goal is to minimize the total cost of transmissions that all nodes have to send, which is also called weighted cost. Following the same idea of our previous work which provided a method based on d-Basis construction to solve cooperative data exchange problem without weighted cost, we present a modified method to solve cooperative data exchange problem with weighted cost. We present a polynomial-time deterministic algorithm to compute the minimum weighted cost and determine the rate vector and the packets that should be used to generate each transmission. By leveraging the connection to Maximum Distance Separable codes, the coefficients of linear combinations of the optimal coding scheme can be efficiently generated. Our algorithm has significantly lower complexity than the state of the art. In particular, we prove that the minimum weighted cost function is a convex function of the total number of transmissions for integer rate cases